The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of. Riemann Surfaces. Front Cover. Lars V. Ahlfors, Leo Sario. Princeton University Press, Jan 1, – Mathematics – pages. A detailed exposition, and proofs, can be found in Ahlfors-Sario , Forster Riemann Surface Meromorphic Function Elliptic Curve Complex Manifold.
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Martins, Damasceno, Awada – Pronto-socorro Pronto-socorro: We have tried, however, to isolate this pq. At the very end of the chapter it is then shown, by essential use of the Jordan curve theorem, that every surface which satisfies the second axiom of countability permits a.
Again, AlHA2 are trivially fulfilled. For instance, a compact space can thus be covered by a finite number of from an arbitrary basis. A topology 9″ 1 is said to be weaker than the topology r 2 if r 1 c r ‘1.
Lars V. Ahlfors, L. Sario – Riemann Surfaces
We proceed to the definition of compact spaces. The definition applies also to subsets in their relative topology, and we can hence apeak of connected and disconnected subsets. For this reason the topological theory of surfaces belongs in this book. Denote the given seta by P. We shall always understand that the topology on a subset is its relative topology. There is a great temptation to bypass the finer deta. For the points of the plane Jl2 we shall frequently use the surfaves notation The sphere 81, also referred to as the u: It follows from Al and A2 that the intersection of an arbitrary collection and the.
This means that points can be identified which were not initially in the same space. The sum of two topological spaces 81, Sa is their union 81 u Bz on which the open seta are those whose intersections with 81 ahlrors Sa are both open.
The empty set and the Whole apace are simultaneously open and closed. On the other hand, it is much easier to obtain superficial ahlfore without use of triangulations, for instance, by the method of singular homology.
The topological product 81 x Sa is defined as follows: A Riemann surface is, in the first place, a surface, and its properties depend to a very great extent on the topological character of the surface. An open set is a neighborhood of any subset, and a set is open if it contains a neighborhood of every point in the set. A closed connected set with more than one point is a conlinuum.
In most a third requirement is added:. B The intersection of any finite collldion of sets in fJI is a union of sets in The section can of course iremann o. In fact, Al – A2 arf’ ouvwusly satisfied; A3 holds if it holds on B. But 01 is also relatively closed in Q. This definition has an obvious generalization to the case of an arbi- trary collection of topological spaces.
Certain characteristic properties which may or may not be present in a topological space are very important not only in the’ general theory, but in particular for the study of surfaces. The chapter closes with the construction of a triangulation. This elementary section has been included for the sakP of l”ompleteness and because beginning analysts are not always well prepared on this point.
This is done by formulating the combinatorial theory as a theory of triangulated surfaces, or polyhedrons. It shows that every open set in a locally connected space is a union of disjoint regions. The empty set and all seta saario only one point are trivially con- nected. It must be observed, however, that the new space is not necessarily a Hausdorff space even if Sis one.
Every open subset of a locally connected space is itself locally connected, for it has a basis conaisting of part of the basis for the whole space. For complete results this derivation must be based on the method of triangulation.
Lars V. Ahlfors, L. Sario – Riemann Surfaces – livro em pdf
Surface nano-architecture of a metalorganic framework Surface nano-architecture of a metalorganic ahlfofs. The main demerit of this approach is that it does not yield complete results.
This is the moat useful form of the definition for a whole category of proofs. If there arc no relations between the points, pure set theory exhausts all poRsibilities.
Then 0 u V p is connected, by 3B, and by the definition of components we obtain 0 u V p c 0 or V p c 0. Since we strive for completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach. It has been found most convenient to base the definition on the consideration of open coverings.
According to usual conventions the union of an empty collection of sets is the empty set 0, and the intersection of an empty collection is the whole space 8.
This reasoning applies equally well to Oz, and we find that 01 and 02 cannot both be nonempty and at the same time disjoint.
Lars V. Ahlfors, L. Sario-Riemann Surfaces
C C X Xcomo The strongest topology is the tliscrete topology in which every subset is open. We shall say that a family of closed seta has the finite inter- M. A apace with more than one point can be topologized in different manners. The construction of a quotient-apace is sometimes preceded by forming the sum of several spaces. Mii1Nr of which ia tloitl. A2 The intersection of any finite coUection of open sets is open.